On Resolving the Column Permutation Ambiguity in the ... - CiteSeerX

2000 Conference on Information Sciences and Systems, Princeton University, March 15-17, 2000

On Resolving the Column Permutation Ambiguity in the Estimates of MIMO System Response Ivan Bradaric, Athina P. Petropulu 1

Electrical and Computer Engineering Department Drexel University, Philadelphia, PA 19104 e-mail: fivan,[email protected]

Abstract | We consider the problem of identifying a n  n MIMO system excited by unobservable non-white and spatially independent inputs. Frequency-domain methods always associate a permutation ambiguity to the system frequency response, which unless computed and compensated for, results in an erroneous solution. We show that, from the permutation-dependent solution we can extract some quantities (invariances) which are independent of the permutation, and which, under certain conditions, can lead to the correct solution.

I. Introduction The blind identi cation of a n  m Multiple-input multipleoutput (MIMO) system is of great importance in many applications, such as communications, biomedical engineering, seismology, etc.. The goal of blind system identi cation is to identify an unknown system H(z), driven by n- unobservable inputs, based on the m system outputs, and subsequently use the system estimate to recover the input signals (sources). In this paper we deal with the case of colored inputs. Most of the existing methods address the problem using higherorder statistics [4], [5], [8], [2]. There are, however, a few methods that under certain conditions, address the problem using second-order statistics only [3], [6], [9]. In [1] a method was proposed that uses frequency domain second-order correlations to recover the system frequency response within a frequency dependent column permutation ambiguity and a frequency dependent diagonal matrix. Under the assumption of unit auto-channels, and for the special case of a 2  2 system, it was shown in [1] that a set of invariants could be extracted from the system estimate, which would suce for the recovery of the cross-channels. In this paper we extend the ideas of [1] to solve the more dicult problem of n  n system identi cation. II. Problem Formulation Let s(k) = [s1 (k)


sn (k)]T be a vector of n statistically independent zero mean stationary sources, H(l) the impulse response matrix with elements fhij (l)g, and x(k) = [x1 (k)


xn (k)]T the vector of observations. Then, the MIMO system output equals:

x(k) =

LX ?1 l=0

H(l)s(k ? l)

(1)

where L is the length of the longest hij (k), and

si (k) = 1

LX g ?1 l=0

gi (l)ei (k ? l)

(2)

This work was supported by NSF under grant MIP-9553227.

Konstantinos I. Diamantaras

Department of Informatics Technological Education Institute of Thessaloniki Sindos, GR-54101, Greece e-mail: [email protected]

where ei (k) is a white signal with unit power, and gi (k); k = 0; ::; Lg ? 1 is the corresponding color. The ultimate goal of blind system estimation/source separation is to estimate the channel matrix and use the estimate to subsequently recover the input sources. To guarantee identi ability it is usually assumed that (A1) Hii (!) = 1; i = 1;


; n. (A2) The product H(z)diagfG1 (z); ::; Gn (z)g is irreducible, where Gi (z) denotes the Z-Transform of gi (k). This condition means that there are no common convolutional terms between gi (k) and hji (k); j = 1;


; n. The frequency response estimate of most existing methods is usually of the form: H^ (!) = H(!)P(!)
(!) (3) where P(!) is a column permutation matrix, and
(!) is a diagonal matrix. In the method of [1], the permutation matrix is function of the frequency (i.e. P(!)), while in the HOSbased methods [4],[8],[2] it is constant. The matrix
(!) is usually of the form:
(!) = diagfG1 (!);


; Gn (!)gdiagfej1(!) ;


; ejn (!)g (4) where i (!) represents phase ambiguity. All Gi (:)'s and i 's are unknown, however, their eect on the channel estimate can be eliminated by enforcing assumption (A2) on H^ (!), i.e., de ning as estimate: H~ (!) = H^ (!)fdiagH^ (!)g?1 (5) Due to the unknown permutation matrix P(!), however, the above estimate is not what we really hoped to get, namely H(!). For example, let us consider the 3  3 case, in where: " 1 H12 (!) H13 (!) # (6) H(!) = H21 (!) 1 H23 (!) H31 (!) H32 (!) 1 and let us assume that for some xed ! the permutation matrix equals: " 0 1 0 # (7) P(!) = 1 0 0 0 0 1 The above form implies that the eect of postmultiplying any matrix by P(!) would be the interchange of the rst and second columns of the matrix. In that case, H~ (!) would equal: " 1 1=H21 (!) H13 (!) # ~H(!) = 1=H12 (!) 1 H23 (!) H31 (!)=H12 (!) H32 (!)=H21 (!) 1 (8)

There are ve more possibilities for H~ (!), depending on the form of P(!). By examining all these possibilities one can see that the only case where the coecients do not contain ratios of mixing lters is when P(!) = I. In [4], [8], where matrix P(!) was not function of the frequency, it was proposed to identify the unknown permutation matrix, by shuing the column of H^ (!), then computing H~ (!) and looking for the H~ (!) that contained coecients with zeros only (no poles). This of course could be done in the case where the mixing lters were all FIR and have no common convolutional components. Such an approach, however, is rather dicult to implement, and obviously does not apply to the case of frequency dependent permutation (e.g. the case of [1]). For the 2  2 case, it was proposed in [1] to extract two invariances from H~ (!) = fH~ ij (!)g, i.e., I12 (!) = H~ 12 (!)=H~ 21 (!) = H12 (!)=H21 (!) (9) 2 ~ ~ I2 (!) = H21 (!) + 1=H12 (!) = H21 (!) + 1=H12 (!)(10) which, under the assumption that the cross-channels did not have common zeros, and each channel did not have zeros in conjugate reciprocal pairs, suce to recover the cross-channels uniquely. In this paper we extend the ideas of [1] to the n  n case, and show that, under some additional assumptions, it is possible to de ne invariances which suce for the reconstruction of all mixing channels. III. Some Definitions Let us assume that in addition to assumptions (A1) and (A2), it holds: (A3) The mixing channels are FIR lters that have no common zeros (A4) The channels hij (n) and hik (n); i 6= j; i 6= k, have no zeros in conjugate reciprocal pairs. For each pair of rows (i1 ; i2 ); i1 ; i2 = 1; :::; n let us de ne the invariances: n n 4 X H~ i1 j (! ) = X Hi1 j (! ) I1n (!; i1 ; i2 ) = ~ H (!) j =1 Hi2 j (! ) j =1 i2 j n n 4 X X H~ i1 j1 (! ) H~ i1 j2 (! ) I2n (!; i1 ; i2 ) = ~ ~ j1 =1 j2 >j1 Hi2 j1 (! ) Hi2 j2 (! ) =

I3n (!; i1 ; i2 )

n X n X Hi1 j1 (!) Hi1 j2 (!)

H (!) Hi2 j2 (!) j1 =1 j2 >j1 i2 j1 n n n 4 X X X H~ i1 j1 (! ) H~ i1 j2 (! ) H~ i1 j3 (! ) = ~ ~ ~ j1 =1 j2 >j1 j3 >j2 Hi2 j1 (! ) Hi2 j2 (! ) Hi2 j3 (! ) n X n X n X Hi1 j1 (!) Hi1 j2 (!) Hi1 j3 (!) = (!) Hi2 j2 (!) Hi2 j3 (!) H j =1 j >j j >j i2 j1

The phase of P n (!; i1 ; i2 ) equals the phase of Inn (i1 ; i2 ) within a linear phase component. The time domain equivalent of P n (!; i1 ; i2 ) is hi1 1 (k) 


 hi1 n (k)  hi2 1 (?k) 


 hi2 n (?k), where \*" denotes convolution. It is well established that a FIR sequence that doesn't contain zero-phase convolutional components can be reconstructed within a scalar from its phase only, even if the phase is known within a linear phase component [7]. Thus, under assumptions (A3) and (A4), Pnn (i1 ; i2 ) can be computed within a scalar constant, i.e, c(i1 ; i2 )2 . Based on the computed P n (!; i1 ; i2 ) and the invariance Inn (!; i1 ; i2 ) we de ne: 4 jP n (! ; i ; i )I n (! ; i ; i )j1=2 Mi1 (!; i1 ; i2 ) = 1 2 n 1 2

= c(i1 ; i2 )

n Y j =1

jHi1 j (!)j

(12)

After these de nitions we are ready to proceed with the algorithm for resolving the permutation ambiguity. IV. Resolving the frequency-dependent Permutation Ambiguity Since in general n  n case the notation becomes dicult to follow, we rst present the 3  3 case. We start with the following invariances: 13 (! ) I13 (!; 1; 2) = H 1(!) + H12 (!) + H H23 (!) 21 H12 (!) + H13 (!) + H12 (!)H13 (!) I23 (!; 1; 2) = H H21 (!)H23 (!) H23 (!) 21 (! ) ~ ~ 12 (! )H13 (! ) I33 (!; 1; 2) = H~ 12 (!)H~ 13 (!) = H H21 (!)H23 (!) H21 (!)H23 (!) H12 (!) + H (!) I13 (!; 1; 3) = H 1(!) + H 13 31 32 (! ) H12 (!)H13 (!) 13 (! ) I23 (!; 1; 3) = H H(!12)H(!) (!) + H H31 (!) + H32 (!) 31 32 12 (! )H13 (! ) I33 (!; 1; 3) = H H31 (!)H32 (!) 21 (! ) I13 (!; 2; 3) = H 1(!) + H23 (!) + H H31 (!) 32 H23 (!) + H21 (!) + H21 (!)H23 (!) I23 (!; 2; 3) = H H31 (!)H32 (!) H31 (!) 32 (! ) H 21 (! )H23 (! ) 3 I3 (!; 2; 3) = H (!)H (!) 31 32 Let us consider the polynomials:

x3 ? I13 (!; 1; 2)x2 + I23 (!; 1; 2)x ? I33 (!; 1; 2) = 0 (13) x3 ? I13 (!; 1; 3)x2 + I23 (!; 1; 3)x ? I33 (!; 1; 3) = 0 (14) 1 2 1 3 2 .. x3 ? I13 (!; 2; 3)x2 + I23 (!; 2; 3)x ? I33 (!; 2; 3) = 0 (15) . n n The roots of the rst, second and third polynomial, respec4 Y H~ i1 j (! ) = Y Hi1 j (! ) Inn (!; i1 ; i2 ) = tively, are: ~ H (!) j =1 Hi2 j (! ) j =1 i2 j H13 (!) ) (16) (X1 (!); X2 (!); X3 (!)) ! ( H 1(!) ; H12 (!); H It is easy to verify that these quantities are the same, whether 21 23 (! ) ~ (!). they are de ned based on H(!) or H 12 (! ) Let us also de ne ) (17) (Y1 (!); Y2 (!); Y3 (!)) ! ( H 1(!) ; H13 (!); H H 31 32 (! ) n Y H21 (!) ) (18) P n (!; i1 ; i2 ) = Hi1 j (!)Hi2 j (!) (11) (Z1 (!); Z2 (!); Z3 (!)) ! ( H 1(!) ; H23 (!); H (!) j =1 32

31

Under assumptions (A3) and (A4), P 3 (!; 1; 2), P 3 (!; 2; 3) and P 3 (!; 3; 1) can be reconstructed within the scalars, c(1; 2)2 , c(2; 3)2 and c(3; 1)2 , respectively, thus we can obtain: M1 (!; 1; 2) = jP 3 (!; 1; 2)I33 (!; 1; 2)j1=2 = c(1; 2)jH12 (!)jjH13 (!)j (19)

kij (!m ; 2; 3) = c(2; 3) ) H21 (!m ) = X (1! ) ; H23 (!m ) = Zj (!m ) (29) i m

kij (!m ; 3; 1) = c(3; 1) ) H31 (!m ) = Y (1! ) ; H32 (!m ) = Z (1! ) (30) i m j m M2 (!; 2; 3) = jP 3 (!; 2; 3)I33 (!; 2; 3)j1=2 Let us now consider the general n  n case. Let us assume = c(2; 3)jH21 (!)jjH23 (!)j (20) that we want to reconstruct an arbitrary channel hi1 i2 , where i1 ; i2 2 f1; 2; :::; ng. We start with the following quantity that M3 (!; 3; 1) = jP 3 (!; 3; 1)I33 (!; 3; 1)j1=2 we are able to obtain as previously discussed: = c(3; 1)jH31 (!)jjH32 (!)j (21) 4 jP n (! ; i ; r)I n (! ; i ; r)j1=2 Mi1 (!; i1 ; r) = 1 1 n Let us consider discrete frequencies !m , i.e., w = 2N !m , n Y and determine the roots X1 (!m ), X2 (!m ), X3 (!m ), Y1 (!m ), = c(i1 ; r) jHi1 j (!)j; r 2 f1; :::; ng(31) Y2 (!m ), Y3 (!m ), Z1 (!m ), Z2 (!m ) and Z3 (!m ) for all !m , j =1 m = 0; 1; :::N ? 1. Furthermore, let us form the following sets of coecients: Let us consider the following polynomial: 4 (1) K (!m ; 1; 2) = fkij (!m ; 1; 2); i; j = 1; 2; 3g; xn ? I1n (!; i1 ; p)xn?1 + I2n (!; i1 ; p)xn?2 ? ::: + (?1)n Inn(!; i1 ; p); p 2 f1; 2; :::; ng (32) kij (!m ; 1; 2) = jXM(!1 (!)mjjY; 1(; !2) )j (22) i m j m and let Xip (!), i = 1; 2; :::n denotes its i-th root. Then it is 4 easy to show that Xip (!) will be equal to one of the following: K (!m ; 2; 3) = fkij (!m ; 2; 3); i; j = 1; 2; 3g; Hi1 1 (!) ; Hi1 2 (!) ; : : : ; Hi1 n (!) (33) kij (!m ; 2; 3) = M2 (!mjZ; 2;(!3)jX)ji (!m )j (23) j m H (!) H (!) H (!) p1

4 fk (! ; 3; 1); i; j = 1; 2; 3g; K (!m ; 3; 1) = ij m kij (!m ; 3; 1) = M3 (!m ; 3; 1)jYi (!m )jjZj (!m )j (24) Let us look closer at the elements of the set K (!m ; 1; 2). By letting Xi (!m ) and Yj (!m ) take all possible values allowed by (16) and (17), and based on the form of M1 (!; 1; 2), it can easily be veri ed that for each frequency !m there will be one and only one combination that will yield an element independent of frequency and equal to c(1; 2), i.e., K (!0 ; 1; 2) \ K (!1 ; 1; 2) \ ::: \ K (!N ?1 ; 1; 2) = fc(1; 2)g (25) Similarly, comparing (16), (18) and (20), and (17), (18) and (21) we conclude: K (!0 ; 2; 3) \ K (!1 ; 2; 3) \ ::: \ K (!N ?1 ; 2; 3) = fc(2; 3)g (26) K (!0 ; 2; 3) \ K (!1 ; 2; 3) \ ::: \ K (!N ?1 ; 2; 3) = fc(3; 1)g (27) It can easily be veri ed that the last three equations can be violated only if one of the following is true: jHij (!)jjHji (!)j = const: jHij (!)jjHjk (!)jjHki(!)j = const:; i 6= j 6= k jHij (!)jjHjk (!)j = const:; i 6= j 6= k jHik (!)j jHij (!)jjHji (!)jjHik (!)jjHki(!)j = const:; i 6= j 6= k However, because of assumptions (A3) and (A4) this is never the case. Based on the obtained constants c(1; 2), c(2; 3) and c(3; 1) (see Eqs. (25)-(27)) we now can recover all cross-channels using the following relations: kij (!m ; 1; 2) = c(1; 2) ) H12 (!m ) = Xi (!m ); H13 (!m ) = Yj (!m ) (28)

p2

pn

Let us now, for previously selected r de ne the following set:

K (!m ; i1 ; r) = fkj1 j2 :::jn (!m ; i1 ; r)g;

m = 0; 1; :::; N ? 1 (34)

where ! = 2N !m , m = 0; :::; N ? 1, and i1 (!m ; i1 ; r) kj1 j2 :::jn (!m ; i1 ; r) = jX 1 (! )jjX 2 (M ; p n j1 m j2 !m )j:::jXjp (!m )j:::jXjn (!m )j j1 ; j2 ; :::jn = 1; 2; :::; n (35) According to (33), and under assumptions (A3) and (A4), one and only one element of this set will not depend on the frequency, corresponding to: Xj11 (!m ) = jjHHi1 1((!!m))jj = jHi1 1 (!m )j 11 m j H ( ! Xj22 (!m ) = jHi1 2(! m))jj = jHi1 2 (!m )j 22 m .. . n Xjn (!m ) = jjHHi1 n((!!m))jj = jHi1 n (!m )j nn m Therefore, for N large enough we have:

K (!0 ; i1 ; r) \ K (!1 ; i1 ; r) \ ::: \ K (!N ?1 ; i1 ; r) = fc(i1 ; r)g (36) After determining the scalar constant c(i1 ; r), Hi1 i2 (!) can be obtained as: kj1 j2 :::jn (!m ; i1 ; r) = c(i1 ; r) ) Hi1 i2 (!m ) = Xjii22 (!m ) (37) Therefore, we were able to recover an arbitrary crosschannel hi1 i2 (k), i1 ; i2 2 f1; 2; :::; ng.

V. Resolving the frequency-independent Let us now assume that: Permutation Ambiguity n jH~ (! )j Y i1 ~j The proposed algorithm for resolving the column permu; ~j = (1) (j ); (1) 6=  (46) ( ! ) = Pi(1) 1 ~ j H tation ambiguity can be simpli ed for the case when the perj =1 j~j (! )j mutation matrix is not the function of the frequency, that is, Then there will be at least 2 mismatches in the Eq.(44), when HOS is implemented instead of second-order statistics. Our approach will be to directly determine a column permuta- that is: tion matrix P and consequently reconstruct all channels. As it will be shown, this time we need only one invariance and P (1) (!) = jHi1 ^~1 (!)j ::: jHi1 k~^1 (!)j ::: jHi1 ~l^1 (!)j ::: jHi1 n^~ (!)j jH1^~1 (!)j jHk1 k~^1 (!)j jHl1 ~l^1 (!)j jHnn^~ (!)j there is no need for determining the roots of the polynomial i1 given by (32). Hi1 k2 (!)j ::: jHi1 l2 (!)j ::: jHi1 n (!)j = jjHHi1 1((!!))jj ::: jjH Let us denote with  a mapping that permutes the columns. 11 k1 k2 (! )j jHl1 l2 (! )j jHnn (! )j For example, one possible mapping for n = 5 is: ; k1 6= k2 ; l1 6= l2 (47)  [35241] [12345] ?! (38) Consequently, ratio given by (45) depends on the frequency. Obviously, any column permutation matrix P has its cor- Therefore, there is one and only one product Pi1 (!) that will responding mapping  that is unique. In addition, for any result in the constant value for the expression given by (45) mapping  we can de ne inverse mapping ?1 . For the ex- and is given by (43). ample given in (38), the inverse mapping is: The channel reconstruction can now be carried as follows. First, we determine the pair of rows (i1 ; i2 ) for which we can ?1 [53142] [12345] ?! (39) determine the quantity given by (42). Next, we form ratio (45) for all possible mappings (n! dierent permutations) and It can easily be shown that for a given column permutation search for one with constant value for all frequencies. The cormatrix P and its corresponding mapping  an element H~ ij (!) responding mapping will determine our column permutation can be expressed as: matrix. In order to illustrate the procedure let us consider the 3  3 H ( ! ) ^ H~ ij (!) = Hij (!) ; ^j = ?1 (j ) (40) case. There are 6 dierent column permutation matrices and j^j consequently, 6 dierent mappings . They are: As it was previously shown, for any pair of rows (i1 ; i2 ) it 1 [123] ?! [123] (48) is possible to uniquely determine the following invariance:

Inn (!; i1 ; i2 ) =

n Y Hi1 j (!)

H (!) j =1 i2 j

(41)

Let us assume that it is possible to nd a pair of rows such that based on phase of Inn (!; i1 ; i2 ) we can recover:

Mi1 (!; i1 ; i2 ) = c(i1 ; i2 )

n Y j =1

jHi1 j (!)j

Let us now form the following product: n 4 Y jH~ i1 ~j (! )j ; ~j = (j ) Pi1 (!) = ~ j =1 jHj~j (! )j According to (40) this product equals:

Pi1 (!) =

jHi1 ^~1 (!)j jH~1^~1 (!)j jH1^~1 (!)j jH~1^~1 (!)j

(42)

=

" 0 0 1 #

jHi1 ^~2 (!)j jHi n^~ (!)j jH~2^~2 (!)j jHn~1n^~ (!)j jH2^~2 (!)j ::: jHnn^~ (!)j jH~2^~2 (!)j jHn~ n^~ (!)j

j =1

jHi1 j (!)j

Therefore, ratio:

4 Mi1 (! ; i1 ; i2 ) = c(i ; i ) Ri1 (!; i1 ; i2 ) = 1 2 P (!) i1

has constant value for all frequencies.

(49) (50) (51) (52) (53)

Let us assume that " 1 H13 (!)=H23 (!) 1=H31 (!) # 1=H12 (!) 1 H21 (!)=H31 (!) H~ (!) = H32 (!)=H12 (!) 1=H23 (!) 1 (43) (54) This corresponds to permutation matrix:

P(!) = P = 1 0 0 0 1 0

(55)

Let us also assume that we are able to recover:

i1 n (! )j = jjHHi1 1((!!))jj jjHHi1 2((!!))jj ::: jjH Hnn (!)j 11 22 n Y

2 [123] ?! [132] 3 [123] ?! [213] 4 [123] ?! [231] 5 [123] ?! [312] 6 [123] ?! [321]

(44)

M1 (!; 1; 2) = c(1; 2)

3 Y

j =1

jH j (!)j 1

(56)

It is easy to verify that using (43), (45) and (48)-(53) we obtain the following 6 dierent ratios: (45) R11 (!; 1; 2) = ~ M1 (!; ~1; 2) jH12 (!)jjH13 (!)j = c(1; 2)jH12 (!)jjH23 (!)jjH31 (!)j (57)

~ ~ R12 (!; 1; 2) = M1 (!; 1~; 2)jH23~(!)jjH32 (!)j jH12 (!)jjH13 (!)j = c(1; 2)jH12 (!)jjH21 (!)j

(58)

~ R13 (!; 1; 2) = M1 (!; 1~; 2)jH21 (!)j jH13 (!)j = c(1; 2)jH13 (!)jjH31 (!)j

(59)

~ ~ R14 (!; 1; 2) = M1 (!; 1; 2)j~H23 (!)jjH31 (!)j jH13 (!)j = c(1; 2)jH13 (!)jjH32 (!)jjH21 (!)j (60) ~ ~ R15 (!; 1; 2) = M1 (!; 1; 2)j~H21 (!)jjH32 (!)j jH12 (!)j = c(1; 2)

(61)

~ R16 (!; 1; 2) = M1 (!~; 1; 2)jH31 (!)j jH12 (!; 1; 2)j = c(1; 2)jH23 (!)jjH32 (!)j (62) R15 (!; 1; 2) being constant implies that the true mapping was: 5 [123] ?! [312] (63) that corresponds to the column permutation matrix given by (55) as expected.

[2] B. Chen and A.P. Petropulu, \Multiple-Input-Multiple-Output Blind System Identi cation Based on Cross-Polyspectra," IEEE Trans. on Signal Processing, submitted in 2000. [3] A. Gorokhov, P. Loubaton, and E. Moulines, \Second Order Blind Equalization in Multiple Input Multiple Output FIR Systems: A Weighted Least Squares Approach", in Proc. ICASSP96, pp. 2415-2418, 1996. [4] Y. Inouye and K. Hirano, \Cumulant-Based Blind Identi cation of Linear Multi-Input-Multi-Output Systems Driven by Colored Inputs", IEEE Trans. on Signal Processing, vol. 45 (6), pp. 1543{1552, June 1997. [5] E. Moreau and J.-C. Pesquet, \Generalized contrasts for multichannel blind deconvolution of linear systems," IEEE Signal Processing Letters, vol. 4, pp. 182-183, June 1997. [6] Lucas Parra, Clay Spence, and B. De Vries, \Convolutive Blind Source Separation based on Multiple Decorrelation", 8th IEEE Workshop on Neural Networks for Signal Processing, Cambridge, UK, September 1998. [7] H. Pozidis and A.P. Petropulu, \Cross-spectrum based blind channel estimation," IEEE Trans. on Signal Processing, vol. 45 (12), pp. 2977-2993, December 1997. [8] S. Shamsunder and G.B. Giannakis, \Multichannel blind signal separation and reconstruction," IEEE Trans. on Speech & Audio Processing, vol. 5(6), pp. 515-527, November 1997. [9] J. Xavier, V. Barroso, and J.M.F. Moura, \Closed Form Blind Identi cation of MIMO Channels", in Proc. ICASSP-98, vol. 6, pp. 3165-3168, Seattle WA, 1998. 1.5

1.5

References [1] K.I. Diamantaras, A.P. Petropulu and B. Chen, \Blind TwoInput-Two-Output FIR Channel Identi cation Based on Frequency Domain Second-Order Statistics," IEEE Trans. on Signal Processing, February 2000.

(k)

13

channel h (k)

23

channel h

channel h32(k)

channel h31(k)

channel h21(k)

channel h12(k)

VI. Preliminary Results 1 1 In this section we include some preliminary simulation re0.5 0.5 sults for the case of 3  3 system with frequency dependent permutations. The cross-channels were taken to be: 0 0 h12 (k) = [1:0000; 0:2998; ?0:1962; 0:0339], −0.5 −0.5 h13 (k) = [1:0000; 0:4557; 0:5338; 0:1477], 1 2 3 4 1 2 3 4 h21 (k) = [1:0000; ?0:5766; 0:6575; ?0:2010], k k 2 1.5 h23 (k) = [1:0000; 0:0450; 0:2000; ?0:1501], h31 (k) = [1:0000; ?0:2420; ?0:1948; 0:2082], 1 1 h32 (k) = [1:0000; 0:5460; 0:8715; 0:5849]. Based on the frequency responses of the cross-channels, ma0.5 trix H~ (!) was computed using (3) and (5), where the permuta0 0 tion matrix P(!) was randomly generated for each frequency. ~ In addition, matrix H(!) was corrupted by the complex Gaus−1 −0.5 1 2 3 4 1 2 3 4 sian noise. k k Although all discrete frequencies can be used in the pro1.5 2 posed algorithm, for some frequencies in a noisy environment it is very dicult to select the right roots using (37) even 1 1 when we have good estimate of the scalar constant. We set 0.5 up a threshold to decide what frequencies are \not good" (fre0 quencies !m for which more than one element of K (!m ; i1 ; r) 0 is very close to the estimated value for c(i1 ; r)) and rejected −0.5 −1 them. 1 2 3 4 1 2 3 4 We performed 100 Monte Carlo runs of the proposed alk k gorithm assuming SNR = 20dB . The estimation results are shown in Figure 1, where the true channels are shown in dash- Figure 1: True vs. estimated cross-channels corresponddot line, the average estimates are shown in solid line and the ing to 100 independent realizations and SNR = 20dB gray area indicates standard deviation.